Question

the venn - diagram below shows the 13 students in ms. wrights class. the diagram shows the memberships for the art club and the dance club. a student from the class is randomly selected. let a denote the event \the student is in the art club.\ let b denote the event \the student is in the dance club.\ the outcomes for the event a are listed in the circle on the left. the outcomes for the event b are listed in the circle on the right. note that josh is outside the circles since he is not a member of either club. (a) find the probabilities of the events below. write each answer as a single fraction. p(a) = 2/13 p(b) = 7/13 p(a and b) = 3/13 p(a|b)= p(b)·p(a|b)=

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

Step2: Substitute given values

We know that $P(A\cap B)=\frac{3}{13}$ and $P(B)=\frac{7}{13}$. So, $P(A|B)=\frac{\frac{3}{13}}{\frac{7}{13}}$.

Step3: Simplify the fraction

When dividing by a fraction, we multiply by its reciprocal. So, $P(A|B)=\frac{3}{13}\times\frac{13}{7}=\frac{3}{7}$.

Step4: Calculate $P(B)\cdot P(A|B)$

We know $P(B)=\frac{7}{13}$ and $P(A|B)=\frac{3}{7}$. Then $P(B)\cdot P(A|B)=\frac{7}{13}\times\frac{3}{7}=\frac{3}{13}$.

Answer:

$P(A|B)=\frac{3}{7}$
$P(B)\cdot P(A|B)=\frac{3}{13}$